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It is common belief that the moon actually does appear larger near the horizon due to a magnification effect caused by the Earth's atmosphere. This is not true. Although the atmosphere does change the color of the moon, it does not magnify or enlarge the moon. In fact, the Moon appears about 1.5% smaller when it is near the horizon than when it is high in the sky, because it is further away by up to 1 Earth radius.

A proof that this effect is an illusion is to establish size relativity in both situations near the horizon and in a high position in the sky. When the moon is near the horizon, holding a small coin at arm's length with one eye closed, positioning it next to the moon and comparing the size of the coin relative to the size of the moon reveals its relative size for that given situation. Then, when the moon is higher in the sky, positioning the coin next to it again, will confirm that the moon looks the same size compared to the coin as when it was near the horizon.

Clouds near the horizon are typically farther away from the viewer, while those high in the sky are closer, giving the impression of a flat, or gently curved, sky surface

Humans may tend to perceive the sky as more or less a surface, but unlike a hemispherical surface, it does not seem to be equally distant from us at all points. When we see clouds, birds and airplanes in the sky, those near the horizon are typically farther away from us than those overhead. If we see an airplane overhead, its image gets smaller and smaller as it nears the horizon. This results in the perception of the sky as a comparatively flat surface. In other words, we perceive the sky near the horizon to be farther away than the sky overhead.

If we perceive the moon to be in the general vicinity of those other things we see in the sky, we would expect it to get equally farther away as it approaches the horizon as well, which should result in a smaller retinal image. But since its retinal image is approximately the same size whether it is near the horizon or not, our brains, attempting to compensate for perspective, assume a low moon must be physically larger. This effect is known as the Ponzo illusion.
This explanation is, of course, the apparent distance theory which now is rejected by modern vision researchers who have specifically researched the moon illusion. See below.

Claims that the apparent size of the moon is affected by the angle of the eyes relative to the head. According to the angle of regard hypothesis, the moon illusion is produced by changes in the position of the eyes in the head accompanying changes in the angle of elevation of the moon.

This hypothesis claims that regardless of eye elevation, the horizon moon will be perceived as being much larger than the zenith moon because the terrain is viewed as a plane extended outward from the observer. Thus, if two objects have the same projected size but appear to lie at different distances from the viewer, the one that seems farther away will look larger.
However, very few people (only about 5%) see the horizon moon as being both larger and farther away. Vision researchers have found that most people see the horizon moon as both larger and closer than the zenith moon! And many people say both moons look about the same distance away. Vision scientists refer to that common disagreement between the apparent distance theory and the available data as the "size distance paradox," and that's why they seek new theories to replace the rejected apparent distance theory. For the latest research do a web search for "moon illusion."

This hypothesis states that the perceived size of an object depends not only on its retinal size, but also on the size of its immediate visual environment. But, relative size can be either relative angular size or relative linear size, or both, and the moon illusion begins as an angular size illusion. For details see link to "the moon illusion explained."
It certainly is a not a size constancy optical illusion for most people.
See the link to "the moon illusion explained."

This hypothesis claims that the illusion is due to the fact that the neural circuitry in our visual system evolves, by neural learning, to a system that makes very efficient interpretations of usual 3D scenes based in the emergence of simplified models in our brain that speed up the interpretation process but give rise to many optical illusions in unusual situations.

Research indicates that 3D vision capabilities emerge and are learned jointly with the planning of movements. After a long process of learning, an internal representation of the world emerges that is essentially well adjusted to the perceived data coming from closer objects. The representation of distant objects near the horizon is less «adequate». In fact, it is not only the Moon that seems larger when we perceive it near the horizon. In a photo of a distant scene, all distant objects are perceived as smaller than when we observe them directly using our vision.

The retinal image is the main source driving vision but what we see is a "virtual" 3D representation of the scene in front of us. We don't see a physical image of the world. We see objects; and the physical world is not itself separated into objects. We see it according to the way our brain organizes it. The names, colors, usual shapes and other information about the things we see pop up instantaneously from our neural circuitry and influence the representation of the scene. We «see» the most relevant information about the elements of the best 3D image that our neural networks can produce. The perceived properties of the objects, such as brightness, angular size, and color, are unconsciously "determined" and are not real physical properties. The illusions arise when the "judgments" implied in the unconscious analysis of the scene are in conflict with reasoned considerations about it.